No, a prism won't split a laser beam into a visible spectrum. Light from most lasers is almost monochromatic, so glass prisms bend the light without splitting it.
However, lasers are not perfect, and their spectrum is not an infinitely narrow line. The "linewidth" for a diode laser is a few megahertz, or about a billionth of a percent of the laser wavelength. So, if you use a prism to bend a laser beam, the beam will become slightly spread out, but only by about a hundred-billionth of the deflection angle.
On the other hand, certain lasers are designed to emit light of several different frequencies at the same time. The argon-ion lasers used for laser light shows are these types. If you use a prism to deflect the beam of this sort of laser, the beam will be split into two or more colors. But once split, each color behaves as an independent monochromatic laser beam.