Blow your own trumpet
Act in a boastful, self-promoting manner.
The term in its present form is 19th century. Anthony Trollope, in his work Australia and new Zealand, 1873 observed:
"In the colonies..when a gentleman sounds his own trumpet he 'blows.'"
That is clearly commenting on what must already have been a well-known phrase, although there aren't known printed citations to support that from before 1873. The use of 'blows' is interesting too, as that matches the current US use as a slang term to describe anything bad or worthless.
Much earlier, there is an example of what appears to be the same meaning as blowing your own trumpet, from Miles Coverdale's The second tome or volume of the Paraphrase of Erasmus upon the Newe Testament, 1549:
"Let them never so muche with their blasyng wordes boaste themselves."
Blasing is no longer used in that context, but in the 16th century is thought to have meant boastful.
The King James Version of the Bible, 1611 also includes a warning against boastfully 'blowing one's own trumpet', although not in those precise words:
6:1 Take heed that ye do not your alms before men, to be seen of them: otherwise ye have no reward of your Father which is in heaven.
6:2 Therefore when thou doest thine alms, do not sound a trumpet before thee, as the hypocrites do in the synagogues and in the streets, that they may have glory of men. Verily I say unto you, They have their reward.
6:3 But when thou doest alms, let not thy left hand know what thy right hand doeth:
6:4 That thine alms may be in secret: and thy Father which seeth in secret himself shall reward thee openly.