Suppose there is a country with Islamic government, if there was an election and 60% of the people voted for a secular government for example, would the Islamic government give up power or not?
Your question reminds me of a poll in Portugal if my memory could help, the physicians there were to vote for "permitting killing fetus in womb, yes or no"? I just remember a physician was too angry, expressing his anger with this analogy: "if everyone on the Earth vote for permitting murder would it then be moral and permitted? Of course not!" I agree with him, not everything is up to us and even if every single person on the Earth but one votes for something to be obligatory or forbidden, that will not always be acceptable and therefore a MUST to that single person not voting for that!
By democracy if you mean a way to govern an Islamic country whose ruler is a prophet or a succeeder of him, introduced in name by the Prophet, then the answer is NO, unless the ruler himself invite the others for a poll:
وَمَا كَانَ لِمُؤْمِنٍ وَلَا مُؤْمِنَةٍ إِذَا قَضَى اللَّـهُ وَرَسُولُهُ أَمْرًا أَن يَكُونَ لَهُمُ الْخِيَرَةُ مِنْ أَمْرِهِمْ ۗ وَمَن يَعْصِ اللَّـهَ وَرَسُولَهُ فَقَدْ ضَلَّ ضَلَالًا مُّبِين
It is not fitting for a Believer, man or woman, when a matter has been decided by Allah and His Messenger to have any option about their decision: if any one disobeys Allah and His Messenger, he is indeed on a clearly wrong Path. (Al-Ahzaab:36)
However, in the present era that neither our prophet SA is directly ruling the Islamic society nor his succeeder introduced in name --Imam Mahdi AS--, democracy can find a place in the overall method of governing an Islamic country. The basic and fundamental rules should always be that of Islam, the overall foreign diplomacy should be determined by Islam, and no poll should be held in which a rule of Allah is proposed to be accepted or not, but after that any rule whose acceptance and rejection has nothing to do with Islam can be proposed to all the inhabitant of that country as poll. This kind of democracy is of course a limited one, if Allah brings the idea the rule should be based on the idea of the creator who knows best, but otherwise there would be a chance that a public decision based on the collected-wisdom be more accurate and trustworthy than a personal-decision based on a personal-wisdom.
Actually note that not every person is eligible to vote for every specific issue and it was better that a governor could give weight to votes of different voters, however, sometimes reserving equal votes for every inhabitant (although not a child which is not eligible to vote in almost any poll!) is the only practical way.