I should get a quest "Bitter Springs Infirmaty Blues" from Lieutenant Markland and "No, Not Much" from Captain Gilles, but the problem is that they don't give me the quests.
I have already done the quest "Climb Ev'ry Mountain". Is this a bug or I'm missing something?

I believe Markland won't give you the quest until Giles mentions No Not Much, since they're connected.

If Giles isn't giving you the quest, perhaps the dialogue options/order you're choosing them in isn't reaching the quest trigger lines - that is, sometimes I've noticed if you don't pick the right 'path' the dialogue will shortly end and you have to start the convo over and pick other options to get to the quest lines. Other than, not sure. The wiki lists several possible bugs (assuming they haven't been fixed by the recent patch) but seems like most interfere with completing, not acquiring.

The only thing to say her now is "Goodbye". So it's bugged then. Thanx for telling this.