Given two real valued orthogonal functions, say $f(x)$ and $g(x)$, if we define an inner product $$ \langle f,g\rangle \ = \ \int_a^b f(x) g(x) dx,$$
which we know satisfies the properties of an inner product, namely $\bf positive \ definiteness$, $\bf linearity$ in the first argument, and $\bf conjugate \ symmetry$.
Is it true that $\langle f,g\rangle = 0$ for any values of $a$ and $b$ such that $a<b$?
Usually not: the choice of $a$ and $b$ is crucial. For instance, consider the orthogonal family of exponential functions $e^{inx}$. Two of these are orthogonal, that is, $$ \langle e^{inx},e^{imx}\rangle = \int_a^b e^{i(n-m)x}\,dx=0\quad(m\neq n) $$ if and only $b-a$ is a multiple of $2\pi$.
Edit: I see now that you asked for real-valued functions. Similar examples work; I chose the exponential functions because they are simple. For instance,the functions $x^n$ and $x^m$, if $n$ and $m$ are nonnegative and $n+m$ is odd, are orthogonal if and only if $b=-a$.