Since we determine the metallicity of a star ([Fe/H] or Z) from surface emission, does this change as it ages? For instance, can a young star with a measured [Fe/H] of -0.02 have a higher value when we hypothetically look at it 10 billion years from now? Do metals dredge up?
If the star is a solar mass or below it will not produce any metals (anything heavier than helium) within 10 billion years of birth. It will be on the main sequence, fusing hydrogen to helium via the pp chain, which does not change the abundance of metals. There is a tendency for the heavier elements it was born with to sink with age, but this is stymied by the outer convective layer that stops this diffusion from occurring, other than to increase the surface helium abundance a little.
If you pick a star that is a little more massive, maybe 1.1 solar masses such that it has reached the asymptotic giant branch after its main sequence and red giant phases, then carbon, oxygen, nitrogen and a variety of much heavier elements are being produced in the interior and mixed to the surface by convection. The heavier elements are produced by "slow" neutron capture (aka the s-process).
Thus asymptotic giant stars can be observed with all sorts of "chemical anomalies". Famously this has included objects where technetium has been detected - this has a short half-life so demonstrates it must have been made inside the star - and also objects like "barium giants" - again, rich in heavy elements produced in the s-process. There are also giant stars that show enhanced carbon or oxygen abundances.
It is unlikely that [Fe/H] would change, because iron is not produced in such stars.
More massive stars do not live for 10 billion years. However, they behave in a more complex way during their shorter lives. Firstly, they burn hydrogen via the CNO catalytic cycle and they have radiative envelopes with little convection. The CNO cycle increases the abundance of nitrogen and this can reveal itself in enhanced surface nitrogen even on the main sequence.
Even more massive stars can lose large fractions of their mass towards the ends of their lives revealing the outcome of previous nuclear burning. These include Wolf-Rayet stars that may show extreme carbon, oxygen or nitrogen enhancements.
Again though, even in more massive stars, the relative iron abundance is unlikely to change because it isn't made in these stars until the final days of their existence.
One metal that does markedly change its abundance during a star's life is lithium. This is "burned" at lower temperatures than hydrogen and is thus depleted in the interiors of even low-mass stars. Various mixing processes then gradually deplete the surface lithium with age. For instance, the Sun has a surface lithium abundance about 100 times smaller than at birth (as judged from the lithium content in meteorites).
Yes, it will
Metals in astronomy, are simply elements that are heavier than hydrogen and helium. From that perspective, even oxygen and carbon are "metals" in a astronomical sense, although they are not chemically "metals".
Basically, assume the Sun as a example. Being a G2V star, the Sun 's core will eventually run out of hydrogen in approx. 7 billion years. This doesn't mean that the hydrogen in the entire star has run out. In fact, the Sun still has plenty of hydrogen left. However, it will be shunted out of the core, and form a shell around the core. This hydrogen shell will now fuse hydrogen at a much faster pace than before, resulting in a much higher temperature, somewhere about 200,000,000 K. This temperature will cause the Sun to expand into a Red Giant, somewhere to more than 200x its diameter, engulfing the Mercury, Venus and (possibly Earth).
Due to this intense temperature, the RGS's electron-degenerate helium core begins fusing rapidly, ramping to about 150,000,000 K before it loses its degeneracy and expands into the star. This helium fuses into carbon via the Triple-$a$ process.
So technically, in a sense, a older star will be more loaded with "metals" than it had when it was young.
However, this is not all.
There is still one thing: Main-sequence.
Main-sequence stars are those stars that fuse hydrogen into helium via the proton-proton fusion reaction, that liberates immense amounts of energy.
It also means that a very old star will still have the same amount of metals as it had when it was young. Your star will not have to be merely old, it also has to quit the main-sequence and enter the asymptotic giant branch (i.e. your star has to stop fusing hydrogen and start munching on heavier elements like helium to produce "metals")!
Red dwarfs and some orange dwarfs (M-type and K-type stars) won't gain any appreciable metallicity over time, as a star has to be atleast 0.5 solar masses before you can fuse helium. Which means that stars below 0.5 solar masses won't make a appreciable amount of "metals" as their core temperatures still fall short of the temperatures and pressures required to fuse helium.
In short, it not only depends on the age of the star, it also depends on whether your star has exited the main-sequence or not, and its mass as well.