I think it might be from the bible. Matthew chapter 6 verse 1-18 Jesus say when you do acts of charity don't announce it with trumpets like the hypocrites do. But I cannot substantiate the origin 100%.
What Is The Origin Of The Phrase Toot Your Own Horn?
I think it's from the Bible at Psalms 75:4-5 where it says '...do not exalt your horn on high...' And the reason given in the surrounding verses says why, 'For God is the judge...this one he abases and that one he exalts..."