Not really. At least for two reasons (but I'm sure there are more). First, Hobbes' "war of all against all" is a specific view of the "state of nature": there is no regulation, no sovereign, no law. Everyone is equal in the sense that everyone is free to take what they can, notwithstanding moral or any other regulative principles. Might is right. Perfect competition, on the other hand, is based on a certain vision of order - market economy. Moreover, perfect competition means there are no monopolies, which is not exactly in accord with "war of all against all" where the strongest is free to take all.
Second, Hobbes' state of nature brings no positive outcome for the community as a whole (well, there is not even a community). Perfect competition, in contrast, supposedly results in a desirable situation where consumers benefit from lower prices and better quality. To put is somewhat differently, in Hobbes' state of nature scarce resources are distributed inefficiently - they just go to the strongest, whereas under the conditions of perfect competition scarce resources go, supposedly, to those who can make the most of them at the lowest price.
All in all, however, the two concepts come from two very different places and I doubt they can be compared meaningfully.