jQuery(document).ready(function() {
  jQuery("#DOPBSPCalendar1").DOPBSPCalendar({
    "calendar": {
      "data": {
        "bookingStop": 0,
        "dateType": 2,
        "language": "en",
        "maxYear": 2020,
        "reinitialize": false,
        "view": false
      },
    },
    "days": {
      "data": {
        "available": [true, true, true, true, true, true, true],
        "first": 7,
        "firstDisplayed": "2019-09-01",
        "morningCheckOut": false,
        "multipleSelect": true
      },
      "text": {
        "names": ["Sunday", "Monday", "Tuesday", "Wednesday", "Thursday", "Friday", "Saturday"],
        "shortNames": ["Su", "Mo", "Tu", "We", "Th", "Fr", "Sa"]
      }
    }

  });
});

var markup = document.documentElement.innerHTML;
var variable = markup.match( /"firstDisplayed"/gi );

var xxx = jQuery("#DOPBSPCalendar1").DOPBSPCalendar({
      ....
});
//xxx JQ объект, в нем и искать средствами же JQ

var xxx = jQuery("#DOPBSPCalendar1").DOPBSPCalendar({
      ....
});
//xxx JQ объект, в нем и искать средствами же JQ

Object(1)
0: div#DOPBSPCalendar1.DOPBSPCalendar-wrapper.notranslate.dopbsp-initialized
accessKey: ""
​​accessKeyLabel: ""
​​align: ""
​​assignedSlot: null
​​attributes: NamedNodeMap [ class="DOPBSPCalendar-wrapper notranslate dopbsp-initialized", id="DOPBSPCalendar1" ]
​​childElementCount: 3
​​childNodes: NodeList(4) [ div.DOPBSPCalendar-container, div.DOPBSPCalendar-sidebar.dopbsp-style4, #text, … ]
​​children: HTMLCollection(3) [ div.DOPBSPCalendar-container, div.DOPBSPCalendar-sidebar.dopbsp-style4, div.DOPBSPCalendar-clear ]
​​classList: DOMTokenList(3) [ "DOPBSPCalendar-wrapper", "notranslate", "dopbsp-initialized" ]
​​className: "DOPBSPCalendar-wrapper notranslate dopbsp-initialized"
​​clientHeight: 1320
​​clientLeft: 0
​​clientTop: 0
​​clientWidth: 860
​​contentEditable: "inherit"
​​contextMenu: null
​​dataset: DOMStringMap(0)
​​dir: ""
​​draggable: false
​​firstChild: <div class="DOPBSPCalendar-container" style="width: 496px;">
​​firstElementChild: <div class="DOPBSPCalendar-container" style="width: 496px;">
​​hidden: false
​​id: "DOPBSPCalendar1"
​​innerHTML: "<div class=\"DOPBSPCalendar-container\" style=\"width: 496px;\">    <div class=\"DOPBSPCalendar-navigation\">        <div class=\"dopbsp-month-year\">September 2019</div>        <a href=\"javascript:void(0)\" class=\"dopbsp-add-btn\" style=\"display: block;\"><span class=\"dopbsp-info\">Add month view</span></a>        <a href=\"javascript:void(0)\" class=\"dopbsp-remove-btn\" style=\"display: block;\"><span class=\"dopbsp-info\">Remove month view</span></a>        <a href=\"javascript:void(0)\" class=\"dopbsp-next-btn\"><span class=\"dopbsp-info\">Next month</span></a>        <a href=\"javascript:void(0)\" class=\"dopbsp-previous-btn\" style=\"display: block;\"><span class=\"dopbsp-info\">Previous month</span></a>        <div class=\"dopbsp-week\">            <div class=\"dopbsp-day\" style=\"width: 69px;\">Su</div>            <div class=\"dopbsp-day\" style=\"width: 69px;\">Mo</div>            <div class=\"dopbsp-day\" style=\"width: 69px;\">Tu</div>            <div class=\"dopbsp-day\" style=\"width: 69px;\">We</div>            <div clas…"
​​innerText: "September 2019\nSu\nMo\nTu\nWe\nTh\nFr\nSa\n \n \n1\n \n \n \n \n2\n \n \n \n \n3\n \n \n \n \n4\n \n \n \n \n5\n \n \n \n \n6\n \n \n \n \n7\n \n \n \n \n8\n \n \n \n \n9\n \n \n \n \n10\n \n \n \n \n11\n \n \n \n \n12\n \n \n \n \n13\n \n \n \n \n14\n \n \n \n \n15\n \n \n \n \n16\n \n \n \n \n17\n \n \n \n \n18\n \n \n \n \n19\n \n \n \n \n20\n \n \n \n \n21\n \n \n \n \n22\n \n \n \n \n23\n \n \n \n \n24\n \n \n \n \n25\n \n \n \n \n26\n \n \n \n \n27\n \n \n \n \n28\n \n \n \n \n29\n \n \n \n \n30\n \n \nOctober 2019\n \n \n29\n \n \n \n \n30\n \n \n \n \n1\n \n \n \n \n2\n \n \n \n \n3\n \n \n \n \n4\n \n \n \n \n5\n \n \n \n \n6\n \n \n \n \n7\n \n \n \n \n8\n \n \n \n \n9\n \n \n \n \n10\n \n \n \n \n11\n \n \n \n \n12\n \n \n \n \n13\n \n \n \n \n14\n \n \n \n \n15\n \n \n \n \n16\n \n \n \n \n17\n \n \n \n \n18\n \n \n \n \n19\n \n \n \n \n20\n \n \n \n \n21\n \n \n \n \n22\n \n \n \n \n23\n \n \n \n \n24\n \n \n \n \n25\n \n \n \n \n26\n \n \n \n \n27\n \n \n \n \n28\n \n \n \n \n29\n \n \n \n \n30\n \n \n \n \n31\n \n \nNovember 2019\n \n \n27\n \n \n \n \n28\n \n \n \n \n29\n \n \n \n \n30\n \n \n \n \n31\n \n \n \n \n1\n \n \n \n \n2\n \n \n \n \n3\n \n \n \n \n4\n \n \n \n \n5\n \n \n \n \n6\n \n \n \n \n7\n \n \n \n \n8\n \n \n \n \n9\n \n \n \n \n10\n \n \n \n \n11\n \n \n \n \n12\n \n \n \n \n13\n \n \n \n \n14\n \n \n \n \n15\n \n \n \n \n16\n \n \n \n \n17\n \n \n \n \n18\n \n \n \n \n19\n \n \n \n \n20\n \n \n \n \n21\n \n \n \n \n22\n \n \n \n \n23\n \n \n \n \n24\n \n \n \n \n25\n \n \n \n \n26\n \n \n \n \n27\n \n \n \n \n28\n \n \n \n \n29\n \n \n \n \n30\n \n \n \n \n1\n \n \n \n \n2\n \n \n \n \n3\n \n \n \n \n4\n \n \n \n \n5\n \n \n \n \n6\n \n \n \n \n7\n \n \n\n\nExtras\nAdults *\n0\n▾\nUnder 23 *\n0\n▾\nYour Booking\nCheck in\t07 September 2019\nCheck out\t15 November 2019\nPrice\t70 ₪\n\n\t\n\nAdults\t0\n\n\nUnder 23\t0\n\n\t\n\nDiscount\tFull trek\n\n- 0 ₪\n\t"
​​isConnected: true
​​isContentEditable: false
​​lang: ""
​​lastChild: <div class="DOPBSPCalendar-clear">
​​lastElementChild: <div class="DOPBSPCalendar-clear">
​​localName: "div"
​​namespaceURI: "http://www.w3.org/1999/xhtml"
​​nextElementSibling: <form class="cart" action="" method="post" enctype="multipart/form-data">
​​nextSibling: #text "
..

jQuery("#DOPBSPCalendar1").data()