Possibly, if your redness is one of the conditions this laser is used for. Ask a provider to name your condition before you ask about the laser.
The clinical document below names the specific conditions a pulsed dye laser is credited with, so start there.
DermNet on the pulse dye laser's track record
DermNet states that for the treatment of port-wine stains, hemangiomas, and facial telangiectasias, the 585 nm pulse dye laser has the best clinical track record for both effectiveness and safety, regardless of patient skin phototype.
Three conditions are named in that sentence, and facial redness in general is not one of them. The phrase "regardless of patient skin phototype" means the statement is not tied to a single skin type. If your redness is not one of the three, ask what the provider would use instead.
Other light-based options and references
These three pages cover light-based treatments for redness. Read an NCBI PMC paper on intense pulsed light for erythematotelangiectatic rosacea, the AAD's page on skin conditions that lasers can treat and Penn Medicine Cosmetic Services' photofacial page for comparison.
The one thing to do before you act
Ask your St. Augustine provider to name the cause of your redness in one sentence. Then ask whether a pulsed dye laser is the device they would use for it, and why.