Not directly. But of course it was the Cold War, so the ultimate option of a nuclear exchange was an implicit subtext to any confrontation between the two sides.
The spur here was a threat from the Soviets to intervene directly (which in turn was not without provocation).
In that letter, Brezhnev began by noting that Israel was continuing to
violate the ceasefire and it posed a challenge to both the U.S. and
USSR. He stressed the need to "implement" the ceasefire resolution and
"invited" the U.S. to join the Soviets "to compel observance of the
cease-fire without delay". He then threatened "I will say it straight
that if you find it impossible to act jointly with us in this matter,
we should be faced with the necessity urgently to consider taking
appropriate steps unilaterally. We cannot allow arbitrariness on the
part of Israel."
The US, as Israel's ally, would feel compelled to intervene directly as well in that case, so this indirectly was a war threat.
What they did was raise the ongoing nuclear readiness posture to DEFCON 3, which is a level at which response forces (not the response itself, just the forces required to carry one out), could be mobilized with only 15 minutes notice. Given the logic of MAD, the biggest danger is being in a position where the other side believes they can catch you flat-footed and destroy you before you can even get a retaliatory strike up. So the higher (lower actually) levels generally are meant to match how imminent an attack from the Soviets might possibly be. They had just threatened to do so, which means a raising of the alert level seemed to be in order.
The extra military activity required for that readiness level was immediately noticeable to the Soviets, which is where that quote came from.