他の車にぶつかりそうになった。
My car was going to bump against another car.

この表現で伝わりますか？

私的感覚では、英語では「私は私の車を他の車にぶつけそうになった。｣と表現するのが適当だと思います。
i almost hit my car against another.
とかでいいと思います。 　

I had a close call when I nearly hit my car against other one. 　

トヨタさんが書いた表現は伝わりますよ。ただもし同じ文を使うなら、Againstを外して下さいね
*My car was going to bump another car.

Res-1さん、Res-2さん、失礼ながら添削させてもらいます。
*I had a close call when I nearly hit another car.
*I had a close call when I nearly bumped another car.

*I almost hit my car against another.

”ぶつかりそう”の程度はケースによって使用する単語が変わってきますが、上記の例文よりややseriousの場合は；（日本語がうまく書けないので、下記から英語で失礼させて下さい）
*I almost smashed into another car.
*I almost crashed into another car.

smash and crash are verbs but can also be used as nouns for example:

* I had a smash in my car
* I had a crash in my car.

Smash and crash describe a stronger event than bump or hit.

”Against” is not a verb and the verbs Smash, crash, bump and hit all describe some type of impact. Against also describes a contact or impact event so it is not necassary to use next to these verbs.BUT if you use it, this not wrong but not natural style usually.