No. This is a very common myth.
People who are having thoughts of suicide are usually relieved to have the chance to talk about their feelings, and this also makes it possible for them to get help. Those who are not suicidal will not start thinking about suicide because they have had a chance to talk about it. However, they might feel more able to talk about suicidal feelings in the future if they have had the opportunity to talk about suicide in a safe environment such as school.
If you would like more information, this video of Professor Anthony Jorm, co-founder and Chair of Mental Health First Aid Australia and an international expert on many aspects of mental health, has some insights.
If you would like to read about some of the research he talks about, read this Evidence Summary by Headspace, Australia’s National Youth Mental Health Foundation.