Is it permissible to speak with a girl—with the intention of marrying her—once every two or three months, merely to check on her, knowing that the conversation does not involve anything unlawful?

Communication between a young man and a non-mahram woman must be cut off due to the presence of enticements to fitna (temptation/strife). The desire for marriage does not justify such communication. If marriage becomes easy, that is good, for "nothing like marriage is seen for two who love one another." But if it is not facilitated, then each should go their own way. In permissible matters like these, it is sunnah to perform Istikhara (prayer for guidance).