I found this term in a scientist book and my effort so far was to copy paste the word into ejje weblio, but weblio didn't gave me an answer. Is this phrase a kind of idiom or something? How can I google something that is suspected to be an idiom?
The literal meaning of 天分 is something like "that which is allotted to you by the heavens". It refers to your natural talents and characteristics, as well as the position in life you're born into, and has a sense of referring to your overall destiny or place in the world.
So to 天分に生きる is to live according to the will of the heavens - to accept and make the most of the nature and role that has naturally been given to you, rather than trying to "change your destiny" and achieve things you may be less well suited for.
I believe it's generally used with fairly positive connotations, based on the idea that not everyone can be destined for great things, and a more humble life can be just as worthwhile as the life of a great person, as long as it's the life that is best for you.
As for how to go about identifying complex expressions like this, looking up the phrase or its individual words in various dictionaries can be a good start. The full phrase 天分に生きる isn't really an idiom itself but a common collocation that naturally proceeds from the meaning of 天分, so in this case you're unlikely to find the full phrase in dictionaries, but looking at the various translations that dictionaries give for 天分 (words I'm seeing on Weblio include "natural gifts", "talents", "destiny" and "mission") can give you a decent sense of the word's connotations.
Looking at the characters themselves can also help, of course - the meaning of 天分 proceeds quite naturally from the kanji it consists of, with 天 referring to "the heavens" (and often by extension "nature" or "the divine") and 分 meaning your "share" or that which has been allotted to you.
If simple Japanese-to-English dictionaries aren't sufficient to fully grasp the meaning of a word, I find resources like Eijiro, which provides a variety of phrasal uses in addition to individual word definitions, can also help to understand how a word is used. In this case, the Weblio page for 天分 already contains some helpful phrasal uses, including the phrase 人は天分を全うすべし, which is very similar in meaning to 天分に生きる.
And of course, if you're comfortable with using them, native Japanese-language resources are often the most insightful of all - Japanese-to-Japanese dictionaries like Goo Jisho will often provide more in-depth definitions than Japanese-to-English resources, and just Googling a word or phrase (optionally followed by とは) will often turn up pages that discuss its origins or at least illustrate how it's actually used. In this case, the very first Google result I found for 天分に生きる was this page, a short article examining the idea, which helpfully begins by defining 天分 in the first paragraph.