"Grow homesick" is not a standard idiomatic expression.
The common idiom related to missing home is "to become homesick." This means to start feeling a longing or sadness for your home and family while you're away from them.
Here are some examples of how it's used:
* "He became homesick after only a week away from his family."
* "She started to feel homesick when she realized how much she missed her friends."
It's possible someone might use "grow homesick" to describe a gradual increase in feelings of longing, but it's not a common or standard phrase.