Can momentum be measured in gram metre/second. If no, then what is the appropriate unit to measure it

No. The appropriate unit is either gram centimeter/second or kilogram meter/second.

Sure. It is called the CGS (Centimeter, Gram, Second) convention. It is used for cases where using meters and kilograms adds extra computation overhead and more significant digits must be carried for equal accuracy.

Example:

What is the momentum of 1 g moving at 1 cm/s
CGS
p = 1 g * 1 cm/s = 1 g-cm/s

MKS (Meter, Kilogram, Second)
Overhead
convert 1 gram to kilograms
1g = 0.001 kg
convert 1 cm to meters
1 cm = 0.01 m
calculation

p = 0.001 kg * 0.01 m/s = 0.00001 kg-m/s (5 significant figures)

Momentum of a body means product of its mass and velocity.So unit of momentum will be the product of the unit of mass and unit of velocity.
So SI unit of mass=kg and SI unit of velocity=m/s
CGS unit of mass =g and CGS unit of velocity=cm/s
So SI unit of momentum=kg m/s.
CGS unit of momentum=g cm/s.
so momentum can be measured in gram metre/second.There is no wrong in it.Both are same.

The SI unit of momentum is Kg m/s
other unit is g cm/s in CGS
and if you multiply SI with 1000 it wil be in g m/s
but it is not good use in calculations

Rarely';It is measured in kgm/s