Can I resolve the krogan / salarian conflict with both sides supporting the Alliance?

Yes you can

To do so, You have to cure the Genophage, and then save the salarian councillor. (Captain Kirrahe or Thane must be alive)

You can also gain both their support under specific circumstances.

1. Wreave must be alive instead of Wrex.
2. You had to destroy Maelon's data in ME2.

With Maelon's data destroyed Eve dies during the run towards the genophage distribution facility, you can then persuade Mordin that with Eve dead Wreave will have no one to control his krogan behavior and he will agree on faking the genophage cure. Wreave thinks he's cured, you get Krogan support. You didn't cure the genophage, you get Salarian support.

If Wrex is alive, the only way is by saving the council dude.

Either way is pretty heartbreaking, You either lose Wrex or you lose Thane or Kirrahe.

spoilers

this is the total number of war assets from the Salarians:

125 from saving the Salarian councilor in me1 
35  for letting Kirahee die and talking to Lt. Tolen at the STG base in me3 
25  from Mordin if you persuade him that curing the genophage is a bad 
      idea with Wreav in charge and Eve dead 
150 from sabotaging the cure and 
8   from a Salarian side mission
= 343

You also get the Krogan too with this

25  from Wreav 
25  from Grunt if he survived me2 and his loyalty mission was completed 
50  from Arlak company 
50  from n7 Cerberus attack 
345 from Krogan clans and also speaking with Diana Allers 
40  from Krogan side mission 
300 Clan Urdnot 
=795

Add them together and you get a total war asset amount of 1138.

When you cure the genophage and lose the extra Salarian support, you get 875+169 which equals to 1044.

So kill Wrex and destroy Maelon's data. All it will do is change dialog, nothing bad.