The connector from the TV that hooks to the coax cable broke off , is there a way to fix it? or how much it may cost me to get it fix from a proffecional (tv is a year old)

It is repairable . I did several but I am not in the TV business. I make a foot long extension cable hanging out the back from where the connector broke off the turner with a new connector on the end.
You will have to call a local TV shop and negotiate a price. It is about a thirty minute job.