A temporary marriage contract cannot be "converted" to a permanent marriage. What would need to be done is the man says to the woman he is in a mutah contract with that he frees her for the remaining period of their contract, and they can then do a permanent marriage contract.
Of course, they would need to agree on a mahr as well, and there is no 'iddah period, because he will be marrying the same person he was with. This means that once he frees her of the remaining mutah period, they can straight away do the permanent marriage.
And Allah knows best.