1. Given lines

   $3x + 4y = 9$ $y = mx + 1$

   We need the abscissa (the $x$-coordinate) of their point of intersection to be an integer.

2. Find the intersection point

   Substitute $y = mx + 1$ into $3x + 4y = 9$:

   $3x + 4(mx+1) = 9$

   $3x + 4mx + 4 = 9$

   $x(3+4m) = 5$

   Hence,

   $x = \frac{5}{3+4m}$

3. Condition for $x$ to be an integer

   Since $m$ is an integer, $3+4m$ is also an integer. For

   $x = \frac{5}{3+4m}$

   to be an integer, $3+4m$ must be an integer divisor of $5$.

   The divisors of $5$ are:

   $\pm 1, \pm 5$

   So we solve:

   $3+4m = 1, -1, 5, -5$

4. Check each case

   • If $3+4m=1$, $4m=-2 \Rightarrow m=-\frac{1}{2}$ Not an integer.

   • If $3+4m=-1$, $4m=-4 \Rightarrow m=-1$ Integer.

   • If $3+4m=5$, $4m=2 \Rightarrow m=\frac{1}{2}$ Not an integer.

   • If $3+4m=-5$, $4m=-8 \Rightarrow m=-2$ Integer.

5. Count integral values of $m$

   The integral values are:

   $m=-1, -2$

   So the number of integral values is:

   $2$

6. Option check

   • A: $1$ ❌
   • B: $2$ ✅
   • C: $3$ ❌
   • D: $0$ ❌

Therefore, the correct option is B.