I'll go the opposite direction to what everyone is saying.The guy who plays Khal Drogo is a natal Leo Sun and he does bring that part of his personality to the role.With a bit of Mars in Gemini also peeking in there.
His whole obsession with bloodline, sons, strength in masculinity and his need to appear strong and dominant(Leo).
I know that, in those days, these traits were necessary, but he could have displayed them in a passive-aggressive manner too.
The Moon in Taurus is his love for all things phsyical and his enchantment with the beautiful Khaleesi.
Most of his interaction with her was in bed and she is able to persuade him, through affection and diplomacy, to do things her way.
Khal Drogo -Leo Sun/Taurus Moon
The Khaleesi actress has natal Sun in Taurus. But she brings the cousin of this to her role-Libra- as she is able to turn enemies into friends and many are enchanted with her charm and beauty(Kahl Drogo included)
The Aquarius Moon is her need to see justice and equality in everything. She is even able to have people follow her without imposing force or threat of death(unlike her husband)
Khaleesi- Libra Sun/Aquarius Moon
Its interesting that he often referred to her as the "Moon to his Sun" because Moon in Aquarius opposes his natal Leo Sun.
At times, she was a able to persuade and convince him to do the opposite of what he intended to do(especially in the love scenes)
She also got him to "feel". And that's the effect of the Moon person on the Sun person: "I want you to "feel" me."
It could also be Libra Sun on Leo Sun. Libras charm and persuasion yield results- even with the ferocious Leo mate.
So I say;
Khal Drogo: Leo Sun/Taurus Moon
Khaleesi: Libra Sun/Aquarius Moon
The square (by sign) between the two moons is indicative of their first meeting. Khaleesi feared him and their first sexual interaction was more like rape.