jmc

We note that $$\frac{17}{4}=4\frac{1}{4}\text{and}\frac{35}{2}=17\frac{1}{2}.$$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 $\boxed{7}.$