That depends on the specifications of the PSU.
6A will be the nominal operating current at rated load i.e. full speed pumping the fluid its designed for against the specified head (back pressure), however, for a brushed motor, the startup current (initially the locked rotor stall current) can be an order of magnitude higher and if mains power is initially applied with the pump connected to the PSU, and the PSU has foldback current limiting, the over-current protection may trip and the pump fail to start. Its also possible to trip the protection switching the 12V DC circuit to the pump, but is less likely as the reservoir capacitors in the PSU will be fully charged and will take the initial surge for a ms or two.
To determine if this is likely to be the case, you need to know the DC resistance of the pump, which lets you calculate the stall current, then consult a detailed datasheet for the PSU. If the PSU has a constant current mode when overloaded, and has enough margin on its output current rating, it will be able to spin up the pump without tripping. I would expect a 10A rated CC/CV PSU, to be satisfactory. However a PC PSU will need a much higher current rating on its 12V output to avoid tripping.
If the pump has a brushless motor, it is likely to have a soft start mode. Consult its datasheet for the peak current during startup.
If you don't have data for a brushless motor pump, then you'll need to power the pump off a low impedance source like a large Lead Acid battery, with a current shunt in the low side, and capture the startup current waveform with a DSO. You can do the same with a brushed motor, but its seldom necessary as you can directly measure its DC resistance.
The other consideration is what are the consequences of the pump supply failing? E.g. if its a sump pump, or a circulation pump aerating a fishpond containing valuable fish, you may want to power it off a deep cycle Lead Acid battery maintained by a float charger that can supply the average operating current for the pump (allowing for its usual duty cycle), with enough Ampere hours capacity to keep it operating while you arrange an alternative supply or alternative pump.