It depends how much you trust your psychologist ... have you ever mention your suicidal thoughts before? If not you may start saying you have sometimes thought about but have no active behaviour or actual plan. Being suicidal is a matter of degrees : there anre people that just think about, or dream about (passive behaviour), some others are looking for informations on how to do that (active behaviour), some others have a plan, some others a datevand method set. The first thing the psychologist has to asset is if you are passive or active, and what stage you are in. If you are passive suicidal usually there is no consequence in saying that, because it is considered low risk - they are just thoughts.
If you want to try, you can start to mention your thoughts and see the reactions. It will help you know if to say any further.
My psychologist know all about my active behaviour (I have almost everything for the SN method) and he told me he cannot stop me or report me, so I freerly speak about.
That said, it is of course up to you.