What is the process by which fluorescent proteins in two photon microscopy are optically stimulated by membrane depolarization?

They are using a GCaMP, specifically GCaMP6s and GCaMP6f.

This is a genetically encoded sensor. Cells expressing the gene make a protein. The protein is inside the cell, in the cytoplasm. This protein includes a domain that can bind calcium ions. There are lots of other proteins in a cell that bind calcium, too, so this is just one of them.

When bound to calcium, the protein is in a different conformation than when not bound to calcium. When a particular wavelength of light shines on the calcium-bound protein, it emits a photon of a different color. Two-photon imaging takes advantage of a summation trick in that two longer-wavelength (reddish) photons of light combine to provide enough energy to release a single shorter-wavelength photon (greenish).

No voltage or energy from the cell is needed to operate the fluorophore, except the energy used to synthesize the protein originally and the small voltage changes that cause calcium channels to open and allow calcium into the cell. It only fluoresces because of the excitation light used in the microscope.

It would be possible that if you had too much calcium binding protein around it would actually bind enough calcium to interfere with cellular signaling. However, people use these proteins quite a bit and it doesn't seem to affect function all that much. Lab mice who have these proteins expressed throughout their brains all their life aren't any behaviorally different than wild type animals.