Can I take a soft suitcase on Qatar Airways?

In these situations the best thing you can do is check the company rules

https://www.qatarairways.com/en-gb/baggage/allowance.html

Baggage allowances may vary according to route and cabin class; please check the exact baggage regulations printed on your ticket.

Baggage guidelines:

Maximum dimension = length + width + height;

A single piece of checked baggage should weigh no more than 32kg (70lb). Any bags weighing more than 32kg (70lb) must be broken down so that each piece weighs less than 32kg (70lb).

So only weight and dimensions are taken into account for your baggage, not materials

Whilst Qatar Airways might not have restrictions on luggage materials, at least one of the airports they fly from/to does:

OR Tambo Airport (aka Johannesburg Airport) in South Africa has recently (25 May 2019) implemented a "no irregular shaped checked baggage" restriction for passengers departing from OR Tambo Airport. This includes bags without at least one flat side and bags with long straps. There is a infographic floating around which also includes soft bags, but I am yet to find the official source so will not link to it just yet.

The reasoning behind it is apparently to do with the airports automated baggage handling systems getting jams from errant bag straps and from bags which continuously roll back on the sections the conveyor belts go uphill.

Passenger baggage will be screened at check-in and passengers with bags that the check-in agents deem not to qualify as "regular shaped" will be guided to the "cling-wrap" stands where for the approx price of ZAR90.00 (about USD6.00) their bag will have a "flat insert" cling-wrapped to the side of it (or will have the long straps cling-wrapped out of the way).

So in answer to your question, Qatar Airways might allow you to fly with a soft suitcase, but depending on the shape the airport you're flying through might not (without a temporary cling-wrap modification).

EDIT: In trying to find the infographic, it appears Dubai Airport implemented the same rule in 2017. Having not flown through either airport since 2017, I cannot personally vouch for these restrictions. But if you think your bag might possibly be classed as "irregular", I'd budget some spare cash for check-in time in case you need to get a "flat insert" cling-wrapped to your bag.