How can curl perform a get request with a data payload?

Above curl request is a valid request, in fact, if you have index and data, then you can check the output of your command.

I tried it in my system and ES index and it gave me proper response.

curl -v -X GET "localhost:9500/querytime/_search?pretty" -H 'Content-Type: application/json' -d'{    "query": {        "query_string" : {            "query" : "(avengers) OR (big apple)",            "default_field" : "movie_name"        }    }}'*   Trying ::1...* TCP_NODELAY set* Connected to localhost (::1) port 9500 (#0)> GET /querytime/_search?pretty HTTP/1.1> Host: localhost:9500> User-Agent: curl/7.64.1> Accept: */*> Content-Type: application/json> Content-Length: 156>* upload completely sent off: 156 out of 156 bytes< HTTP/1.1 200 OK< content-type: application/json; charset=UTF-8< content-length: 905<{  "took" : 4,  "timed_out" : false,  "_shards" : {    "total" : 1,    "successful" : 1,    "skipped" : 0,    "failed" : 0  },  "hits" : {    "total" : {      "value" : 3,      "relation" : "eq"    },    "max_score" : 0.14874382,    "hits" : [      {        "_index" : "querytime",        "_type" : "_doc",        "_id" : "1",        "_score" : 0.14874382,        "_source" : {          "movie_name" : "Avengers: Infinity War"        }      }    ]  }}

As mentioned in the official manual of curl command, if you are using *nix based system, then you can search below in the manual of curl.

-G, --get When used, this option will make all data specified with -d, --data, --data-binary or --data-urlencode to be used in an HTTP GET request instead of the POST request that otherwise would be used. The data will be appended to the URL with a '?' separator

As explained in this SO answer, it also depends on the web-server to parse the body in GET request.