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The fact of Yitro's conversion passes transparently in the Torah commentary.
However, I wondered why he assumed he'll be accepted to the Jewish people and not rejected? Especially for those who claim he came AFTER the Matan Torah, so only the actual participants of the Maamad might be considered "the Jews".
Yisro's daughter Tzipora was in Egypt (as explained by the Ramban quoting Midrash Shemos Rabba here Why Moses didn't care for his sons after leaving Egypt? ) with Moshe Rabeinu when he went as an Emmisary on behalf of Hashem to let the Jewish people go. When he arrived in Egypt and was reminded of how he escaped from Pharoh, he called his son Eliezer which means G-d aided me. Shemos 4,20 says that they both went to Egypt*:
ויקח משה את אשתו ואת בניו וירכבם על החמר וישב ארצה מצרים
When the Jews were commanded to do the Korban Pesach it says Shemos 12,48:
וכי יגור אתך גר ועשה פסח לה' המול לו כל זכר ואז יקרב לעשתו והיה כאזרח הארץ.
When the convert converts and lives among you and makes a Paschal lamb he shall first circumcise and then he can bring it.
The Mechilta says on that Parshas that it apply for all future generations:
זאת חוקת הפסח בפסח מצרים ובפסח דורות הכתוב מדבר, דברי רבי יאשיה
So when Tzipora went back to Midian she probably told her father Yisro that he could convert to Judaism after she told him about how they went out of Egypt.
*See Rashi however who says that when they were about to arrive in Egypt Aharon told Tzipora to return. According to this Yisro might not have known about whether converts were accepted but he still pushed to try and convert just like Ruth did after she was initially rejected by Naomi. This shows how genuine a convert Yisro was!
