as a therapist, I concur with the other commenters. your therapist may have you create a safety plan (safe people, places, ways to make environment safer, coping skills, etc) which may end in something to the effect of “if you are experiencing any imminent risk of harm to yourself or others please go to/be transported to the nearest emergency department for appropriate care.” otherwise you can just collaborate more effectively together with your suicidal ideation now out in the open.
I’ve told both my therapist and psychiatrist when I have sh urges and suicidal thoughts (but no plan). And usually they want to talk about it more or adjust my medications. This was also a fear of mine, bc I don’t want to be hospitalized again. But my therapist reassured me that the only time she would have to report me is if I had an active plan to kms and she believed I was a danger to myself or others.