Is it possible to have multiple Conjugate Priors? In Bayesian probability theory, can a probability distribution have more than one conjugate prior for the same model parameters? 

I know that the Normal distribution has another Normal distribution or the Normal-Inverse-Gamma distribution has conjugate priors, depending on what the model parameters are.

If multiple priors exists, then kindly cite some. If multiple priors do not exist, what is the proof that it cannot exist?
I am not 100% sure I understand your question, but consider a parameter space $\Theta$ with prior probability $f(\theta)$ and a probability distribution $p(x|\theta)$ over a space $X$ indexed by this parameter. The conjugate prior is simply obtained by computing Bayes rule:

\begin{equation*}
p(\theta|x) = \dfrac{f(\theta) p(x|\theta)}{\int_{\theta' \in \Theta}{f(\theta') p(x'|\theta)}}
\end{equation*}

Therefore, if $\theta$ (the model parameter) is given, the posterior distribution $p(\theta|\cdot)$ is unique and given this formula. Or, in other words, if $p(\theta|x)$ and $q(\theta|x)$ are two conjugate priors for $p(x|\theta)$, it is obvious that $p(\theta|x)=q(\theta|x)$ for any $x$.
I am not 100% sure I understand your question, but consider a parameter space $\Theta$ with prior probability $f(\theta)$ and a probability distribution $p(x|\theta)$ over a space $X$ indexed by this parameter. The conjugate prior is simply obtained by computing Bayes rule:

\begin{equation*}
p(\theta|x) = \dfrac{f(\theta) p(x|\theta)}{\int_{\theta' \in \Theta}{f(\theta') p(x'|\theta)}}
\end{equation*}

Therefore, if $\theta$ (the model parameter) is given, the posterior distribution $p(\theta|\cdot)$ is unique and given this formula. Or, in other words, if $p(\theta|x)$ and $q(\theta|x)$ are two conjugate priors for $p(x|\theta)$, it is obvious that $p(\theta|x)=q(\theta|x)$ for any $x$.