How to induce a steady increase in cyctoplasmic [Ca2+] in HEK293T cells? And is there any simple marker/method to confirm it?
As far as I know, microscopy (be it with dyes like fluor or with genome-encoded calcium indicators) is really your only good bet, especially for getting quantitative sensitivity, which I think is what you're looking for. Changes in cytosolic calcium are incredibly dynamic both in time and space, which is something to consider when asking how calcium affects your two proteins.

I do know of an example where investigators transferred cells to buffers with different Ca2+ concentrations and permeablized them with a mild detergent to raise calcium levels (Borges-Pereira et al 2015). I'm not sure if that would work for your system, but it's worth taking a look, as I think it raises the levels pretty gently and steadily, and you have a better idea of what your intracellular concentration is going to be without directly visualizing it. Considering you won't be able to tell the Ca2+ concentration at any given time without microscopy, things like SOCE and intracellular chelation rates might be important, so you'll want to watch out for when after treatment you start your assay, perhaps trying a few different time points

Hope that was of some help!