strike up (something)
To initiate, instigate, or begin something. I saw you striking up a conversation with Jack—what did you two discuss? You might be able to strike up a deal with her if you mention your father's connection to the business. The detective and the con man struck up a unique friendship.
strike up a friendship
to become friends (with someone). I struck up a friendship with John while we were on a business trip together. If you're lonely, you should go out and try to strike up a friendship with someone you like.