Being a multilingual person myself, I think it depends on the person. The correct answer would probably be "not necessarily"
I speak four languages, and think in all four - well, I haven't been speaking Catalan for some time, so neither have I been thinking in that language.
I happen to be a bit of a "day dreamer", with a lot of imagination. So I can momentarily forget that I am originally English-speaking. For instance, if I am driving my car to meet with a Spanish speaking person, then I find myself thinking in Spanish. the same with French. But I think it is possible to be bilingual or multilingual without having that particular characteristic.
During the last 40 years I have lived either in French or in Spanish, with the exception of one year when I was living and working mostly in English. However I find when I am counting or adding up figures in my head, if I am tired or if there is the slightest difficulty, I will be thinking of the numbers in English. (Maths is my weak point).