Green’s Dictionary of Slang

1. (orig. US) to play music with great energy and enthusiasm.

1954		Louis Armstrong Satchmo 23: I first heard Buddy Bolden play [...] He was blowing up a storm [W&F].
1957		Shake, Rattle and Roll [film script] Say, Fats, a bunch of the kids would like to listen you blowin’ up a storm.
1960, 1975		Wentworth & Flexner DAS.

2. (US drugs) to get highly intoxicated from smoking a large quantity of marijuana.

1958

W. Motley Let No Man Write My Epitaph (1959) 121: They lit the real thing and, free in the country, blew up a storm, laughing, talking, talking silly, feeling great.

3. to make a fuss.

1965

A. James America’s Homosexual Underground 79: Little snot nose blows up a storm.