NO...this is a PUBLIC form that everyone from Argentina to Hijackistan can see.
IF the Computer let you post your phone number, you would be contacted by a variety of fraudsters and criminals "pretending" to be TTX support and trying to steal your personal info.
You only contact TTX by calling them using the proper procedure here:
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https://ttlc.intuit.com/community/using-turbotax/help/what-is-the-turbotax-phone-number/00/25632
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Or by asking for specific information in this forum, and waiting for someone to reply...but without posting personal ID info..