strike up a conversation

strike up a conversation (with sb) (start a conversation, usually with sb you don’t know too well) — завязать разговор: разговориться (контекст.)

Example 1: Jane's attempt to strike up a conversation with the old woman was much more successful than Brian's. They really hit it off right away. — Джейн гораздо лучше, чем Брайану, удалось завязать разговор со старушкой. Они сразу понравились друг другу; тут же поладили.

make conversation (to talk to someone that you do not know well, in order to be polite, not because you really want to talk to them; to talk or converse for the sake of conversing, with no particular purpose, and usually under some social compulsion) -- говорить, чтобы что-нибудь сказать; перебрасываться словами, вести пустой, бессодержательный разговор; вести светский разговор, вести светскую беседу; вести незначащий разговор (из вежливости, для соблюдения приличий)

If you make conversation, you cause someone to talk to you, esp. about unimportant matters.
This type of conversation is also called small talk

.....he was only making conversation while they waited for the train
...."Nice party, isn't it?" I said, trying to make conversation.
.....I was just trying to make conversation.