Can there be oxygen dissolved in water in a hydrogen atmosphere?
This question is very good as it brings together many disciplines$-$astronomy, chemistry, atmospheric physics. Regretfully, as a consequence it is hard to compentently answer by a member of one field. I look forward to a healthy discussion in the comment section.



Hydrogen

A scientifically sound story ought to be such in most aspects. Specifically, 


how does that planet possess hydrogen in high concentrations?


Hydrogen is a molecule with small molecular mass. As such it occupies the high end of Maxwell-Boltzmann speed distribution:

$$f(v)\begin{align}&amp;=4\pi\sqrt{{\left(\frac M{2\pi RT}\right)}^3}v^2\exp\left(-\frac{M{v^2}}{2RT}\right) 
\end{align}.$$

$\ce{H2}$ would thus escape the atmosphere over time since they often reach escape velocity

$$v_e\approx\sqrt{\frac{2GM_{planet}}{r}}.$$ 

Possible explanations why this would not have occured include:

1. Relatively young planet

A newly-formed celestial body is very likely to be hot. As such, it will not contain liquid water, definitely not to the extent of forming oceans.

One study by Tian et al. 'A Hydrogen-Rich Early Atmosphere' ($2005$) suggest the possibility of oceans and a concurrent hydrogen mixing ratio of $30$%. However, this claim has been contested.


  Tian et al.’s argument that hydrogen escape was inefficient on early Earth assumes that the upper atmosphere would be cold as on Venus, because the atmosphere is composed mostly of CO2. Their claim is controversial because their model depends on gases other than hydrogen to provide the radiative cooling, but their model does not actually include gases other than hydrogen. In particular, the key assumption that the upper atmosphere was cold is not obvious and needs to be addressed quantitatively. [----] 
  
  Visconti’s calculations contrast markedly with the cold thermospheres in Tian et al.’s pure hydrogen escape models.
  
  source: Zahne et al. 'Earth’s Earliest Atmospheres'. (2010) link


2. Massive size and a magnetoshpere

Massive size would mean it has not really lost any atmosphere. Unless it is close to a star, in which case hydrodynamic escape would become important. These planets are mostly gas giants. We do not really know what is going on at their center, but it is almost certainly not a liquid ocean.

Without a magnetosphere the planet's atmosphere would be washed away by solar wind, similarly to that of Mars.

3. Hydrogen-producing microorganisms, e.g., a hyperactive, modified Chlamydomonas reinhardtii

This would be your best  bet. I am not qualified to discuss the likelihood of simultaneous thriving of both hydrogen-producing and oxygen-producing organisms. At face value$-$and this is speculation$-$it is probably not viable. Especially with so high pressures of hydrogen (high solubility).

What if?

Whether the reaction between $\ce{O2}$ and $\ce{H2}$ actually takes place (with considerable speed) depends on numerous variables. The ratio of these gases, temperature, pressure.


  Can there be oxygen dissolved in water in a hydrogen atmosphere?


Short answer is yes.


  [Is it] a significant amount?


Probably not. There ought to be some equilibrium amount but negligible. Even leaving aside the reaction between hydrogen and oxygen, the oxygen would be used to oxidise minerals. After these are, in a sense, saturated and when aquatic organisms adapt to more oxygen-rich water will the concentration of $\ce{O2}$ grow. It is now that the high hydrogen content becomes an issue.

Extra discussion requires actual calculation with supplied parameters, and the definition of what constitutes a significant amount. I have provided some sources for the kinetics of $\ce{O2/H2}$ mixture if this a route you wish to undertake.


Marcus Ó Conaire et al.. 'A Comprehensive Modeling Study of Hydrogen Oxidation' link
Alekseev et al.. 'The effect of temperature on the adiabatic burning velocities of diluted hydrogen flames: A kinetic study using an updated mechanism' Combustion and Flame, 162, 5, 1884$-$1892. (2015). link
Martin Hersch. 'Hydrogen-Oxygen Chemical Reaction Kinetics in Rocket Engine Combustion'. NASA Technical Note. (1967). link


Rogue planets

In a paper by David J. Stevenson 'Life-sustaining planets in interstellar space' (Nature vol. 400, page 32, 1999, link) he proposes that planets which form sufficiently far from their star or are gravitationally accreted may be sent on a course outside their star system. These planets could retain their hydrogen because the effective temperature would be fairly low: estimated to be about $30 \ \mathrm{K}$. According to Stevenson, 


  [P]ressure-induced far-infrared opacity of $\ce{H2}$ may prevent these bodies from eliminating internal radioactive heat except by developing an extensive adiabatic (with no loss or gain of heat) convective atmosphere.


Thus surface temperatures may be enough for liquid water to exist. Assuming the planet collided with enough asteroids and comets at a stage where water was retained we arrive at the conditions:


hydrogen-rich atmosphere, Stevenson estimates that the pressure could be $100$ to $10000\ \mathrm{atm}$, so your intuitive estimate was pretty adequeate;


Sidenote: Pressure at the bottom of oceans on Earth is (at the deepest) about $1000\ \mathrm{atm}$. So we would have the same amount of pressure on the surface of the ocean.


liquid oceans.


A possibility for life would be anaerobic chemoautotrophs: they do not need oxygen, and would be able to be their own source of organic molecules. They do require a geothermal energy source. Volcanic activity at the bottom of the ocean floor suffices. Through random genetic mutation these organisms might transition from being obligatory to being facultative anaerobs. Remember: if $\ce{O2}$ is your product, $\ce{CO2}$ is likely the raw material the organism is utilising.

I have no idea how things could proceed from here. We could have a cosy ecosystem in such an ocean (especially near vents) but how and why there would be evolutionary pressure toward oxygen-rich life at other depths... I simply do not know. Only way is to decrease hydrogen content but, well, we need that to keep warm :-). Perhaps a gradual enrichment or replacement of hydrogen by other gases without lowering temperature further is possible? (see suggestion below)

As Stevenson notes, 


  If life can develop and be sustained without sunlight (but with other energysources, plausibly volcanism or lightning in this instance), these bodies may provide a long-lived, stable environment for life (albeit one where the temperatures slowly decline on a billion-year timescale). The complexity and biomass may be low because the energy source will be small, but it is conceivable that these are the most common sites of life in the Universe.


Suggestion

If you wish for more information, consider follow-up questions on our


biology,
physics,
world building,
astronomy site.


I suggest against asking in all sites at once. Consider taking a concrete point (for example the transition from anaerobic to aerobic on biology.SE). Read their guidelines to be sure it is on-topic, and check for possible duplicates. Link to this question when asking again.

Good luck!