Does oxygen gas form hydrogen bonds?
To make it clear, the question is: do both molecules have to contain hydrogen in order for hydrogen bonding to occur?


No, as noted by Prakhar in his answer, tertiary amines are one class of compounds that can participate in hydrogen bonding (as an acceptor) when there is no hydrogen atom directly bound to the participating atom.

As an example, trimethylamine can hydrogen bond just fine with water:



More information can be found, e.g., here at ChemGuide (scroll down to "Solubility in water").