To unlock the quest to fight Redmaw you must first proceed through the Hunters Guild quests which has a number of steps.
1) Unlock a minimum 3 half-suns in any of trials at the Hunting Grounds. Unless you fail the trial by running out of machines you should get a half-sun since the time limit is 20 minutes.
2) Go to the Hunters Lodge in Meridian and speak with first Ligan, then Ahsis , then finally Telanah. She will send you out to collect 6 trophies: 3 from Sawtooths, 2 from Ravagers, and 1 from a Stalker. They will be automatically in the loot gained from felled machines until you have the number needed, so don't worry about farming for drops. Return them to Ahsis.
3)Talanah will send you to meet her in a nearby town being attacked by Glinthawks. When you arrive you must fight off the Glinthawks before being sent to see what is drawing them to the area. You will eventually find the source and need to acquire a Snapmaw heart to end the danger.
4)Talanah will send you out for more hunting trophies, this time 1 each from a Stormbird and a Thunderjaw. When you return to the lodge to turn them in you will be told that Talanah and Ahsis are out hunting Redmaw.
This unlocks the Redmaw quest. First you must go to the area designated by the map to the far south of Meridian. There you will have to fight
Follow Talanah until you need to use your focus to find the trail, then follow the glowing trail straight to Ahsis and Redmaw.
Redmaw is a Thunderjaw with higher health and strength than normal. Take him on as you would any other Thunderjaw by tearing off his components and attacking his weak points while dodging his attacks. If he charges Talanah and leaves you a moment to breath, pick up his disk launcher to do massive damage. Eventually you will fell the legendary machine.
Speak to Ahsis, and then return to the Hunters Lodge. Speak with Talanah and the Trophy will pop.