We note that $\frac{17}{4}=4 \frac{1}{4}$ and $\frac{35}{2}=17 \frac{1}{2}$. Therefore, the integers between these two numbers are the integers from 5 to 17, inclusive. The odd integers in this range are $5,7,9,11,13,15$, and 17, of which there are 7.