"A call on a river" is a fixed phrase. It means that someone who has the legal right to a certain amount of water from a river can make "a call on a river" to exert their claim to their legal rights to that water--even if such a claim can "inconvenience" other people or entities that have lesser (junior) rights.
Basically it is to make a legal claim to the amount of river water you are entitled to. Note I'm not an expert on the subject. You can Google "call on the river" and get many examples of usage.
This excerpt may help explain the phrase
Under Colorado’s prior appropriation system, when a senior rights holder places a call on a river, upstream junior appropriations must stop diverting water until the senior right has its full allocation.
“That’s their legal rights, and they’re not doing anything wrong,” McClow said about calls by rights holders. Historically, he said, the sharing of water for irrigation worked much more effectively.
In your example, the city of Fort Collins has the senior claim to water rights, so in times when less water is available (as during a drought, or when water levels fall to a certain point, or whatever situation is described in your source) and Fort Collins is not getting its legally alloted amount of water, it can put a "call on the river" to exert its legal authority to the water. When it does that, entities with junior (lesser) water rights have to stop using the water or stop restricting the water from being available to Fort Collins. So,
the North Poudre Irrigation Co. is required by the Office of the Colorado State Engineer to cease groundwater pumping operations since they are damaging the senior surface water rights owned by the City of Ft. Collins.