TLDR: No, the cost-benefit wouldn't work out, for reasons that I'll get to in a second. However some other parties to the conflict may, in part, have incentives of the kind the question suggests.
First of all, the vast majority of energy commodities are traded via contracts, usually arranged well ahead of time. The reason to arrange ahead of time is because energy commodities are notorious for both melt-ups and melt-downs, thus both buyer and seller have excellent motivation to reduce the risk profile.
Contracts can be over exchanges, via futures and options (basically highly standardized contracts with very open pricing mechanisms), or in the case of the energy world, quite often via direct contracts between buyers and sellers. The latter fall into several categories when it comes to the pricing scheme : fixed price (often for one-off tenders, i.e. a shipload on such-and-such a date), or in the case of long-term recurring arrangements or pipeline situations, they're quite often hub-indexed (i.e. track a strip of natgas futures on a particular exchange to reflect the buyer's locality), or oil-indexed (same but with energy-equivalent oil index). The hub-indexed and oil-indexed structures effectively introduce a moving-average filter into the price paid, something like order of 6-12 months timescale.
The point of this digression is that the price impact of the currently quite elevated European (TTF hub) natural gas price, has only a limited impact on the actual prices paid for the majority of LNG cargo today.
However, if the situation were to persist over a longer period of time, of course a seller with long-term contracts would eventually cash in.
A final natural gas market technicality is worth mentioning because of the way it affects incentives for the US. Due to limited LNG export capacity, North America's natural gas price zone is effectively isolated from the rest of the world. Thus we have prices in the $3-5 / MMBtu range in North America, while it is, at this moment, around $25 / MMBtu in Europe, and perhaps a little less in East Asia. The interesting dynamic arising from this is that, increases to global LNG demand don't affect North American natural gas price, when the transmission mechanism (export terminals) is maxed out. North American exporters (the owners of the export terminals) see the full swings in global price, but most North American consumers and producers see only a fraction of it.
Aside from that, the common motivational context is that the "benefit" of crisis pricing would go to all sellers of natural gas, not just Russia.
The point here is that if, for example, a significant natural gas exporter such as Norway or Indonesia (to use some neutral examples who to be clear would never do such a thing) were in a position to create a 6-month-long public perception of a crisis that would threaten the energy supply (but not their own), they would then benefit handsomely.
Now for the reasons against the proposition, which are unique to Russia:
First, Russia tends to have a larger share of its sales at fixed-price arrangements compared to some competitors.
Next, in the case of Russia, is the threat of economic sanctions. These are largely at the discretion of the US, who is a rival natural gas exporter, in a position to benefit from a price rise! This is in contrast the EU, Russia's primary natural gas customer, with a long-term relationship fixed by pipelines. Unlike the US, EU is quite prone to negative side effects, should they be forced to cut off their biggest energy supplier. However, the choice may not be theirs -- past experience has showed that US often applies sanctions unilaterally. Moreover, different EU members have different overriding geopolitical priorities (e.g. fear of Russia, or they have an LNG terminal on their territory). This adds up to an environment where Russia is likely to face sanctions if a war breaks out, even if the net effect of the sanctions is more destructive to EU than to Russia.
Next is the Nord Stream II pipeline, although this has arguably already been nearly mooted by strong signals from EU earlier this year that they wish to permanently delay its commissioning. But if there is any hope for Russia of bringing the pipeline into service, it would certainly be killed by a war.
Lastly, Russia enjoys extensive cooperation from European multinationals (think firms like Siemens, but more specialized for building gas terminals and processing plants, oil services and so forth). This cooperation would go away, and their host countries would react to any crisis by attempting to diversify their long-term energy-infrastructure-building partnerships elsewhere - i.e. to the seaborne market via Qatar, Australia, US, or to developing LNG exporters in Africa or South America.
So on the whole, Russia would experience some partial benefit in the portion of its energy exports whose format (LNG vs pipe) or contract terms (hub-indexed) facilitate profit. However, the inconvenience of undercutting its main customer, and potentially creating a crisis that would get out of hand, makes the proposition illogical.
However, it's possible to speculate about other parties, who wish to create some of the negative outcomes, and are in a position to create the perception of an imminent military crisis.
That's not to say it is the primary motivation for any party, but more like if there were to be, say, a vote in a legislature or parliament on a relevant national action coming as a result of the crisis, then legislative members with large energy exporting firms among their major donors, might perhaps see things in a particular way.