I farmed 19 alexandrites from fates yesterday... 133 fates from start to finish.
Among those there was a few (less than 5 I think) that I only got silver on, but I'm not sure if gold is required to get an alexandrites so I count them all.
This was with a relic weapon animus equipped.
What weapon were you using?
Edit:
I stayed mainly in low level areas like Western Thanalan, Central Thanalan, Middle La Noscea during my farming. Tried some other areas too, but I found those the best as the FATEs go down really quick.
I used the counter under achievement Destiny's Child (Battle -> Battle) to keep track of how many FATEs I completed.
Since I'm still below 3000 completed FATEs, that works fine.