Question

If a fluid moves in constant velocity could it have constant pressure?

Answer 1

Your questions itself explains why so. The velocity in the fluid is a point property, it can vary from point to point in the flow. In the steady flow, the velocity is constant at a point and it is invariant with respect to time,it doesn't mean that the velocity is constant every where in the fluid flow(if it is constant everywhere then it would be called an uniform flow). The velocity can vary due to area change,body forces etc. Temporal invariance at a point is the only characteristic of a steady flow.