There is some misleading information in the literature about this issue, due (as mentioned by Henrik Winther) to special properties of compact groups.
Suppose $\pi$ is an irreducible representation of a real group.
If $\pi$ is self-dual then it supports an invariant bilinear form, which is symmetric or skew-symmetric. Which one is given by the Frobenius Schur indicator $\epsilon(\pi)=\pm1$.
If $\pi$ is self-conjugate, it is real or quaternionic. Write the real/quaternionic indicator $\delta(\pi)=\pm1$ respectively.
If $\pi$ is Hermitian (admits an invariant Hermitian form, not necessarily positive definite), then self-dual and self-conjugate are equivalent (a stronger statement than the first comment above). If $\pi$ is furthermore unitary then $\delta(\pi)=\epsilon(\pi)$.
There is an elementary formula for $\epsilon$ when $\pi$ is finite dimensional. See Bourbaki, Lie Groups and Lie Algebras, Chapter 7-9, or see The Real Chevalley Involution, arXiv:1203.1901 for a simpler proof.
Proposition: $\epsilon(\pi)=\chi_\pi(\exp(2\pi i\rho^\vee))$
where $\chi_\pi$ is the central character and $\rho^\vee$ is one-half the sum of the positive coroots. Obviously this is independent of the real form. In general, however, $\delta(\pi)$ is sensitive to the real form.
Corollary: If $\pi$ is unitary, in particular if $G$ is compact, the real-quaternionic indicator is $\delta(\pi)=\chi_\pi(\exp(2\pi i\rho^\vee))$.
This gives the simplest answer to the original question, assuming it was only about compact groups. For non-compact groups the relationship is more complicated, although there is still a closed formula; this is the subject of a forthcoming dissertation by Ran Cui at the University of Maryland.