Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.
1. **Analyzing the conditions for \( p \):**
   - Given \( p + 6 \) is a prime number, \( p \) cannot be divisible by 3. If \( p \) were divisible by 3, then \( p + 6 \) would be divisible by 3 and greater than 3, hence not prime.
   - Given \( p + 10 \) is a prime number, \( p \) cannot be equivalent to 2 modulo 3. If \( p \equiv 2 \pmod{3} \), then \( p + 10 \equiv 2 + 1 \equiv 0 \pmod{3} \), making \( p + 10 \) divisible by 3 and greater than 3, hence not prime.
   - Therefore, \( p \equiv 1 \pmod{3} \).

2. **Analyzing the conditions for \( q \):**
   - Given \( q + 4 \) is a prime number, \( q \) cannot be equivalent to 2 modulo 3. If \( q \equiv 2 \pmod{3} \), then \( q + 4 \equiv 2 + 1 \equiv 0 \pmod{3} \), making \( q + 4 \) divisible by 3 and greater than 3, hence not prime.
   - Given \( p + q + 1 \) is a prime number, \( q \) cannot be equivalent to 1 modulo 3. If \( q \equiv 1 \pmod{3} \), then \( p + q + 1 \equiv 1 + 1 + 1 \equiv 0 \pmod{3} \), making \( p + q + 1 \) divisible by 3 and greater than 3, hence not prime.
   - Therefore, \( q \equiv 0 \pmod{3} \). Since \( q \) is a prime number, the only possibility is \( q = 3 \).

3. **Finding suitable \( p \):**
   - We need to find \( p \) such that \( p, p + 6, p + 10, p + q + 1 = p + 4 \), and \( p + 10 \) are all prime numbers.
   - Since \( q = 3 \), we need \( p, p + 4, p + 6, p + 10 \) to be prime numbers.

4. **Checking possible values of \( p \):**
   - We need to check \( p \) such that \( p \equiv 1 \pmod{3} \) and all the numbers \( p, p + 4, p + 6, p + 10 \) are prime.
   - Let's check the values:
     - \( p = 7 \):
       - \( p + 4 = 11 \) (prime)
       - \( p + 6 = 13 \) (prime)
       - \( p + 10 = 17 \) (prime)
     - \( p = 13 \):
       - \( p + 4 = 17 \) (prime)
       - \( p + 6 = 19 \) (prime)
       - \( p + 10 = 23 \) (prime)
     - \( p = 37 \):
       - \( p + 4 = 41 \) (prime)
       - \( p + 6 = 43 \) (prime)
       - \( p + 10 = 47 \) (prime)
     - \( p = 97 \):
       - \( p + 4 = 101 \) (prime)
       - \( p + 6 = 103 \) (prime)
       - \( p + 10 = 107 \) (prime)

5. **Conclusion:**
   - The pairs \((p, q)\) that satisfy the conditions are \((7, 3)\), \((13, 3)\), \((37, 3)\), and \((97, 3)\).

The final answer is \( \boxed{ (7, 3) } \), \((13, 3)\), \((37, 3)\), \((97, 3)\).