On some speakers the grill part, not the bezel or frame of the grill assembly, but the actual grill with the small holes in it is pressed into the bezel/frame.
You take a tool with a point small enough to fit in the holes, or even a very small screwdriver, and you gently pry the grill up and out of the bezel/frame. You want to try and get a hold of the side of the grill just before it enters the bezel. You need to be careful and not try to pull to much out at once and bend the grill. Work your way out in both directions from one corner until you can get that corner free.
On my JBL control 5 this is what i have to do, and your grill material look exactly the same as my JBL's. There is a pliable mastic in the groove that "holds" the grill in place, once you get one corner up you can work it little by little around the grill.
If this works then there should be mounting screws visible to remove the bezel, or the speaker may be attached to the bezel and the whole unit comes out.
To reassemble you just push the grill back into the groove in the bezel.
I do not know if this how your speaker is constructed so proceed with caution.
All that said, If you can not figure it out, you can use painters tape and carefully tape around the bezel, paint and remove the tape before the paint dries.