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 a 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 $\epsilon$. 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.

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.