Swift as the wind, quiet as the forest, fierce as fire, immovable as the mountain.

Yes. The four images are Sun Tzu's, in The Art of War, chapter VII, sections 17 and 18, even though the line is usually handed to Takeda Shingen. The Chinese is 其疾如风，其徐如林，侵掠如火，不动如山. Giles translates rapidity like the wind, compactness like the forest, raiding and plundering like fire, and immovability like a mountain. The popular English — swift, quiet, fierce, immovable — is this passage in shorter wording. The sentences are Sun Tzu's. The banner only abbreviates them.

Shingen's banner abbreviated the four clauses to four characters, 風林火山, read Fūrinkazan: wind, forest, fire, mountain. The shortening is his. The sentences are not. "Quiet as the forest" is looser than Giles, who says compactness, and "fierce as fire" drops the setting Giles keeps, which is raiding and plundering. Cite Sun Tzu, VII.17–18, Lionel Giles, 1910. Mention the banner as his abbreviation of those lines, not as the source. Do not cite Takeda Shingen as the author of the words.