I received a government grant from a European country to write a book. Do I need to report this grant on my tax return and pay taxes off it?

If you are a US citizen or resident you must report all world-wide income from any source, unless there is an express exclusion in the tax code. Off the top of my head I cannot think of why that would be excluded unless you are a student and the grant allows you study or research. See https://www.irs.gov/taxtopics/tc421

There may be something else, though. So don't take that as a certainty. 

Also, if you paid foreign tax on the grant, you should be able to claim a foreign tax credit for the amount of that tax (so you aren't taxed twice). If so answer the TT foreign tax credit interview questions and ask here if you have any trouble.

BTW income an any "ascension to wealth" (I.e. something that makes you richer, so therefore getting cash for a loan is not income because your debt offsets the cash received).