結論

自分の環境の場合はDocker経由で対処できた。
原因はよく分からん。

docker run --rm -t -v $PWD:/app fukamachi/qlot install

おまけ

一応、エラーを載せておく。
ユーザー名やパスは伏字に加工していない。
動作環境はWindows11 + WSL2 + Ubuntu22.04.1 LSTです。

moga@DESKTOP-DTS0P1Q:~/pg/my-game$ qlot install
Installing Quicklisp to /home/moga/pg/my-game/.qlot/ ...
Unhandled SB-BSD-SOCKETS:OPERATION-TIMEOUT-ERROR in thread #<SB-THREAD:THREAD "main thread" RUNNING
                                                              {10045F0123}>:
  Socket error in "connect": ETIMEDOUT (Connection timed out)

Backtrace for: #<SB-THREAD:THREAD "main thread" RUNNING {10045F0123}>
0: (SB-DEBUG::DEBUGGER-DISABLED-HOOK #<SB-BSD-SOCKETS:OPERATION-TIMEOUT-ERROR {10034205C3}> #<unused argument> :QUIT T)
1: (SB-DEBUG::RUN-HOOK SB-EXT:*INVOKE-DEBUGGER-HOOK* #<SB-BSD-SOCKETS:OPERATION-TIMEOUT-ERROR {10034205C3}>)
2: (INVOKE-DEBUGGER #<SB-BSD-SOCKETS:OPERATION-TIMEOUT-ERROR {10034205C3}>)
3: (ERROR SB-BSD-SOCKETS:OPERATION-TIMEOUT-ERROR :ERRNO 110 :SYSCALL "connect")
4: (SB-BSD-SOCKETS:SOCKET-ERROR "connect" 110)
5: (SB-BSD-SOCKETS::CALL-WITH-SOCKET-ADDR #<SB-BSD-SOCKETS:INET-SOCKET 0.0.0.0:58740, fd: 5 {100340DA43}> (#(205 251 199 144) 80) #<FUNCTION (FLET SB-BSD-SOCKETS::WITH-SOCKET-ADDR-THUNK :IN SB-BSD-SOCKETS:SOCKET-CONNECT) {7FCE1B7FEA2B}>)
6: ((:METHOD SB-BSD-SOCKETS:SOCKET-CONNECT (SB-BSD-SOCKETS:SOCKET)) #<SB-BSD-SOCKETS:INET-SOCKET 0.0.0.0:58740, fd: 5 {100340DA43}> #(205 251 199 144) 80) [fast-method]
7: ((:METHOD QL-NETWORK::%OPEN-CONNECTION (QL-IMPL:SBCL T T)) #<unused argument> "beta.quicklisp.org" 80) [fast-method]
8: ((:METHOD QL-NETWORK::%CALL-WITH-CONNECTION (T T T T)) #<unused argument> "beta.quicklisp.org" 80 #<FUNCTION (LAMBDA (QL-HTTP::CONNECTION) :IN QL-HTTP:HTTP-FETCH) {100303587B}>) [fast-method]
9: (QL-HTTP:HTTP-FETCH #<QL-HTTP:URL "http://beta.quicklisp.org/quicklisp.lisp"> #P"/tmp/qlot-9KMPQRQM/quicklisp-DZDG9H5X.lisp" :FOLLOW-REDIRECTS T :QUIETLY NIL :IF-EXISTS :RENAME-AND-DELETE :MAXIMUM-REDIRECTS NIL)
10: (QLOT/PROXY::QLOT-HTTP-FETCH "http://beta.quicklisp.org/quicklisp.lisp" #P"/tmp/qlot-9KMPQRQM/quicklisp-DZDG9H5X.lisp")
11: (QLOT/INSTALL/QUICKLISP::FETCH-INSTALLER #P"/tmp/qlot-9KMPQRQM/")
12: (QLOT/INSTALL/QUICKLISP:INSTALL-QUICKLISP #P"/home/moga/pg/my-game/.qlot/")
13: (QLOT/INSTALL:INSTALL-QLFILE #P"/home/moga/pg/my-game/qlfile" :QUICKLISP-HOME NIL :INSTALL-DEPS T :CACHE-DIRECTORY NIL)
14: (QLOT-COMMAND "install")
15: (SB-INT:SIMPLE-EVAL-IN-LEXENV (APPLY (QUOTE MAIN) ROSWELL:*ARGV*) #<NULL-LEXENV>)
16: (SB-INT:SIMPLE-EVAL-IN-LEXENV (ROSWELL:QUIT (APPLY (QUOTE MAIN) ROSWELL:*ARGV*)) #<NULL-LEXENV>)
17: (SB-EXT:EVAL-TLF (ROSWELL:QUIT (APPLY (QUOTE MAIN) ROSWELL:*ARGV*)) NIL NIL)
18: ((LABELS SB-FASL::EVAL-FORM :IN SB-INT:LOAD-AS-SOURCE) (ROSWELL:QUIT (APPLY (QUOTE MAIN) ROSWELL:*ARGV*)) NIL)
19: (SB-INT:LOAD-AS-SOURCE #<CONCATENATED-STREAM :STREAMS NIL {10028B5023}> :VERBOSE NIL :PRINT NIL :CONTEXT "loading")
20: ((LABELS SB-FASL::LOAD-STREAM-1 :IN LOAD) #<CONCATENATED-STREAM :STREAMS NIL {10028B5023}> NIL)
21: (SB-FASL::CALL-WITH-LOAD-BINDINGS #<FUNCTION (LABELS SB-FASL::LOAD-STREAM-1 :IN LOAD) {7FCE1B7FF6CB}> #<CONCATENATED-STREAM :STREAMS NIL {10028B5023}> NIL #<CONCATENATED-STREAM :STREAMS NIL {10028B5023}>)
22: (LOAD #<CONCATENATED-STREAM :STREAMS NIL {10028B5023}> :VERBOSE NIL :PRINT NIL :IF-DOES-NOT-EXIST :ERROR :EXTERNAL-FORMAT :DEFAULT)
23: ((FLET ROSWELL::BODY :IN ROSWELL:SCRIPT) #<SB-SYS:FD-STREAM for "file /home/moga/.roswell/bin/qlot" {10028B0F23}>)
24: (ROSWELL:SCRIPT "/home/moga/.roswell/bin/qlot" "install")
25: (ROSWELL:RUN ((:EVAL "(ros:quicklisp)") (:SCRIPT "/home/moga/.roswell/bin/qlot" "install") (:QUIT NIL)))
26: (SB-INT:SIMPLE-EVAL-IN-LEXENV (ROSWELL:RUN (QUOTE ((:EVAL "(ros:quicklisp)") (:SCRIPT "/home/moga/.roswell/bin/qlot" "install") (:QUIT NIL)))) #<NULL-LEXENV>)
27: (EVAL (ROSWELL:RUN (QUOTE ((:EVAL "(ros:quicklisp)") (:SCRIPT "/home/moga/.roswell/bin/qlot" "install") (:QUIT NIL)))))
28: (SB-IMPL::PROCESS-EVAL/LOAD-OPTIONS ((:EVAL . "(progn #-ros.init(cl:load \"/etc/roswell/init.lisp\"))") (:EVAL . "(ros:run '((:eval\"(ros:quicklisp)\")(:script \"/home/moga/.roswell/bin/qlot\"\"install\")(:quit ())))")))
29: (SB-IMPL::TOPLEVEL-INIT)
30: ((FLET SB-UNIX::BODY :IN SB-IMPL::START-LISP))
31: ((FLET "WITHOUT-INTERRUPTS-BODY-3" :IN SB-IMPL::START-LISP))
32: (SB-IMPL::%START-LISP)