Solutions
Persons: N, B, C, D, S, M, G and H.
Months: January, February, March, April, May, June, July and August.
Steps:
1) G sits third to the right of the person whose exam is in May.
2) The person whose exam is in August sits second to the right of G.
Case 1:

Case 2:

Case 3:

3) Neither B nor S has their exams in either May or August.
4) B and S are immediate neighbors of each other.
5) Neither B nor S is an immediate neighbor of G.
Case 2 is not possible.
Case 1:

Case 3:

6) H sits third to the right of the person whose exam is in January.
7) Neither B nor S has exams in January.
8) H’s exam is not in August.
Case 1:

Case 3:

9) Only two people sit between S and the person whose exam is in July.
10) The person, whose exam is in February sits on the immediate left of D.
11) Only one person sits between S and N.
The above conditions are not satisfied in case 1.

12) C’s exam is before July.
13) S’s exam is after April.
14) G’s exam is after N’s exam.
15) S’s exam is after April means her exam should be in June.
16) C’s exam should be in May.
17) G’s exam is after N’s exam means B’s exam is in March and G’s exam is in April.
The final arrangement will be as shown:

Hence, two persons are sitting between C and the person whose exam is in April.