I am a male Muslim. If her mother (a Muslim) approves and even requests that I spend some time with a her daughter who is alone in another country, is it permissible for me to check up on the daughter as to respect wishes of her mother?
it is encouraged to check up on her and make sure that she is well and to help her if she needed assistance especially if it is a foreign land and there is no Muslim community. and it is better that you help than another one because her mother asked you and she is Muslim (if a random person asked you for help you would help the same thing apply for Muslim girls) and there is a reward for helping people.
but keep your interaction limited do not overstep like starting an intimate conversation or make it a daily routine to meet for no reason (do not become best friends) you can say you can interact with her like your old lady neighbor who sometimes needs help.
in the future, if you want get married and you are ready to get married and you like her and you feel you both suit each other call her parents and tell them your intention and ask if you can know her better and see their response.