Whatever translation you use, and whether 'pneuma' has a capital 'S' or not, and whether springs of living water refer to the Holy Spirit or not, the point of that verse is the location of the Lamb of God. It is because of where he is that he can feed the innumerable great crowd of people, and give them living water.
Where, then, is the Lamb located? He is in the centre of the throne of the Almighty - he is in the midst of it. (That point is also stated in Rev.5:6 & 22:1-3 making a three-fold witness.)
Further, two verses previously we learn that this multitude of people out of all nations and tribes are "before the throne of God, and serve him day and night in his temple, and he that sitteth on the throne shall dwell among them" (verse 15). Does not the Lamb dwell amongst his people, the people of God, and that by the Holy Spirit? Yet the single verse you ask about does not actually make that point, in itself.
There is clearly a trinitarian link between the Almighty sitting on that throne, and the glorified Son of God (the Lamb) in the centre (midst) of that one throne (for those with eyes to see it). But to find the link between God, Christ and the Holy Spirit in the book of the Revelation, other verses need to be invoked.
Go back to where that one glorious throne is described in chapter four. One aspect of the description is "and there were seven lamps of fire burning before the throne, which are the seven spirits of God" (verse 5). Then, in chapter five, the Lamb makes his first appearance where the description of him includes, "a Lamb as it had been slain, having seven horns and seven eyes, which are the seven spirits of God sent forth into all the earth" (verse 6)
The number seven represents heavenly perfection, so the throne of the Almighty is perfectly guarded by the surrounding seven-fold spirit of God, which also acts as the Lamb's (symbolic) seven eyes. And that is the seven-fold spirit of God now going through the earth since Jesus was gloriously resurrected and made his return to heaven. There you have Revelation's clear connections between the Almighty's throne, and the Lamb's position in the midst of that throne, and the Holy Spirit's guarding and all-seeing aspects in regard to both God and the Lamb. Nobody can break that triple link. They can dismiss that link if they like, but the Holy Spirit guards and sees and freely moves. The symbolic aspects of Revelation's descriptions teach as much about the Holy Spirit as it teaches about God and Christ. But Rev.7:17 does not have "the trinity in view" as you ask. Read the rest of it for that.