I would start by removing Q1 and replacing it with a resistor, say about 1k. See if the circuit gives stable readings. It's possible that you are picking up flickering in your light source.
Also, do you have a decoupling cap across the op-amp power pins? Usually a .1uF. You might also need about 10uf along with that to smooth the supply. Both caps should be very close to the chip.
After that, a cap across the 10k resistor in the feedback (220pf might be a decent value), maybe even up to .1uf to slow the response and eliminate oscillations.
Op amp circuit layout can be very critical. Many schematics drawings are perfect, but how the parts are laid out will ruin the design. A picture of your project might help a lot.