This is the quickest way to do this.
Memorise the following table which contains the atomic number of all the noble gases. Noble gases are placed at the end of every period, so if you remember this table, then using a bit of common sense, we can easily locate the position of any element.
1st period : helium - 2
2nd period : neon-10
3rd period : Argon- 18
4th period : Krypton -36
5th period : Xenon -54
6th period : Radon - 86
Now you are good to go.
Example 1) Say you want to find the atomic number of an element placed in group 4th and period 3.
Neon(10) is the last element of period 2. Come down to the next period ,i.e period 3. We need to reach four blocks to the right to get to group 4. Going each block to the right will increase the atomic number by 1. Therefore, going 4 blocks to the right will give us the atomic number = 10+4= 14.
Example 2) 4th period : 17th group
The last element of period 3 is Argon(18). To get to 4th period and 17th group, we need to come down to period 4 and go 17 blocks to the right. The required atomic number is 18+17=35.
Note : This also requires you to remember that there are 2 elements in the first period, 8 elements each in the second and third period, 18 elements each in third and fourth period.