Как закрасить дату в QCalendarWidget по нажатию кнопки

Попробуйте так:

from PyQt5.QtWidgets import QApplication, QWidget, QPushButton, QCalendarWidget 
from PyQt5.QtCore    import Qt, QRectF, QDate
from PyQt5.QtGui     import QPainter, QColor, QFont, QBrush


class CalendarWidget(QCalendarWidget):                                         # +++
    def __init__(self, parent=None):                                           # +++
        super().__init__(parent)                                               # +++
        self.parent = parent                                                   # +++
       
    def paintCell(self, painter, rect, date):
        painter.setRenderHint(QPainter.Antialiasing, True)
        if date == self.parent.date:                                           # self.parent.date
            painter.save()
            self.brush = QBrush(Qt.Dense6Pattern)  
            self.brush.setColor(Qt.red)
            painter.setBrush(self.brush)
            painter.drawRect(rect)
            
            painter.setPen(QColor(168, 34, 3))
            painter.setFont(QFont('Decorative', 10))            
            painter.drawText(QRectF(rect), Qt.TextSingleLine|Qt.AlignCenter, str(date.day()))
            painter.drawText(rect, Qt.AlignCenter, 'Hello\n\nWorld') 
            painter.restore()
        else:
            QCalendarWidget.paintCell(self, painter, rect, date)


class Ui_Form(object):
    def setupUi(self, Form):
        self.resize(750, 650)
        
        # Календарь
#        self.Calendar = QCalendarWidget(Form)                                 # ---
        self.Calendar = CalendarWidget(Form)                                   # +++
        
        self.Calendar.setGeometry(3, 10, 740, 550)
        self.Calendar.setStyleSheet("QCalendarWidget QTableView{\n""font-size: 30px;\n""/*color: white;\n""background-color: transparent;*/\n""}")
        self.Calendar.setObjectName("calendarWidget")
        # Кнопка
        self.SelectDate = QPushButton(Form)
        self.SelectDate.setGeometry(250, 580, 220, 60)
        self.SelectDate.setStyleSheet("font-size: 40px;\n""color: white;\n""background: lightgreen;")
        self.SelectDate.setText("Отметить")


class Logic(QWidget, Ui_Form):
    date = None                                                                # +++
    
    def __init__(self):
        super().__init__()
        self.setupUi(self)
   
        self.SelectDate.clicked.connect(lambda: self.fillDate())

    def fillDate(self):
        self.date = self.Calendar.selectedDate()
        
        self.Calendar.updateCells()                                            # +++


if __name__ == "__main__":
    import sys                                  
    app = QApplication(sys.argv)    
    window = Logic()                           
    window.show()
    sys.exit(app.exec_())