Are the precious metals (silver, gold, platinum) present everywhere throughout the Earth's crust at least in the trace quantities?
The average concentration of gold in Earth's crust is about 2 ppb (±0.5, depending who you're asking). This is about 2 milligrams Au per tonne of rock. It's not much, but it still equals about 6×1018 atoms, or 6 billion billion atoms.

Seawater has on average 0.004 ppb of gold. This equals 4 micrograms Au per tonne of seawater. Much less, but still about 1×1016 atoms of gold, or 10 million billion atoms.

So to answer your question:


  ...contain fewer than one atom of gold per cubic meter or tonne of
  their mass


The answer is a clear no.



Edit

These seemingly huge numbers do not mean that there is a lot of gold in the rock. That 0.004 ppb of gold which means 10 million billion atoms, is still only 0.0000000004% of the rock. This is clearly a completely insignificant amount of gold. Nonetheless, this insanely tiny amount of gold still an insanely large amount of atoms of gold. There are a lot of atoms out there.