How to Calculate the amount of a investment ? For example 1300 at 4% compounded quarterly for 6 years

Whenever they have a compounding interval qualifier, that means the rate expressed is a nominal rate. Nominal means in name only and is often used for a value that's wrong but close enough.

This is because 

The practice of annualizing a rate by multiplying it by the number of intervals in a year is mathematically wrong but was in use well before we figured out algebra. When they say 4% per annum compounded quarterly, they are saying that it is 1% per quarter which is actually 4.06% per year not 4% per year as there are 4 quarters in a year and 1.01^4 = 1.0406.

Your wording is 

A little bit ambiguous, are you looking for what investment would become $1,300 after six years or what a $1,300 investment will become after six years.

If it took six years (24 quarters) to build up to $1,300 at 4% compounded quarterly (1% per quarter) then the original investment is:

P = $1,300 / 1.01^24
.:
P = $1,023.84

Now if you invested $1,300 at 4% per annum compounded quarterly for six years, you would have a balance of:

FV = $1,300 * 1.01^24
.:
FV = $1,650.66